Continuous Wavelets and Frames on Stratified Lie Groups I

Abstract

Let G be a stratified Lie group and L be the sub-Laplacian on G. Let \(0 \neq f\in \mathcal{S}(\mathbb{R}^+).\) We show that Lf(L)δ, the distribution kernel of the operator Lf(L), is an admissible function on G. It is always in the Schwartz space; one can choose f so that it has all moments vanishing, or has compact support with arbitrarily many moments vanishing. We also show that, if ξ f(ξ) satisfies Daubechies' criterion, then L f(L)δ generates a frame for any sufficiently fine latticesubgroup of G. Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (again assuming that the lattice subgroup is sufficiently fine). In particular, if the dilation parameter is 21/3, and the lattice subgroup is sufficiently fine, then the "Mexican hat" wavelet, Le-L/2δ, generates a wavelet frame, for which the ratio of the optimal frame bounds is 1.0000 to four significant digits.